Scattered locally C*-algebras

In this paper, we introduce the notion of a scattered locally $C^{\ast }$ -algebra and we give conditions for a locally $C^{\ast }$-algebra to be scattered. Given an action $\alpha$ of a locally compact group $G$ on a scattered locally $C^{\ast }$-algebra $A[\tau_{\Gamma }]$, it is natural to ask under what conditions the crossed product $A[\tau_{\Gamma }]\times_{\alpha }G$ is also scattered. We obtain some results concerning this question.